Nontrivial weak solutions for nonlocal nonhomogeneous elliptic problems

This article concerns the following class of nonlocal nonhomogeneous elliptic problems: −∫Ωg(x,u)dxβdiv⁡(a(x)∇u)=(g(x,u))αin Ω,u=0on ∂Ω, where Ω is a bounded domain of Rn(n≥1), α, β∈R, a(x) is a matrix with variable coefficients and g:Ω×R→R is a positive function which are defined almost everywhere with respect to the variable x. By using the Schauder and Tychonoff fixed-point theorems, we establish two existence theorems of weak solutions for this problem according to two bundles of hypotheses on α, β and g.